A compact and efficient high-order gas-kinetic scheme

Abstract In this paper, a compact fifth-order gas-kinetic scheme (GKS) is proposed to solve the Euler equations of compressible flows. It is based on the framework of a one-stage efficient high-order GKS (EHGKS), which can achieve arbitrary high-order accuracy in both space and time, and the newly-developed compact reconstruction technique, combining both the ideas of the simple weighted essentially non-oscillatory scheme (WENO) and Hermite WENO (HWENO). The evolution model of EHGKS provides the time-dependent variables at cell interfaces, which can directly compute the cell-average slope. Thus the necessary information for a fifth-order compact HWENO reconstruction technique is obtained, and also making it possible to improve simple WENO as a compact reconstruction by determining the quartic polynomial based on the compact stencil. The one-stage time discretization technique of EHGKS makes it possible to utilize the reconstruction only once in each computing time step, which is very different from the present compact multi-stage GKS schemes. The newly-developed compact reconstruction provides the polynomial, not only the pointwise variables but also their spatial derivatives for the interface variables evaluations directly. Several numerical examples are given to validate the accuracy and effectiveness of the proposed compact one-stage scheme. And the numerical results also demonstrate that the new scheme is more efficient than the original EHGKS method with WENO reconstruction technique.